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      【NOI2019模拟赛（五十三）】流浪
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  <ul class="article-tag-list"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/动态规划-DP/">动态规划(DP)</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/生成函数/">生成函数</a></li></ul>

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        <h3 id="题意简述"><a class="markdownIt-Anchor" href="#题意简述"></a> 题意简述</h3>
<p>你位于平面直角坐标系的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span>处，每次可以在四个方向选一个方向走过去，一共走<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span></span></span></span>步</p>
<a id="more"></a>
<p>对于一条路径，它对答案的贡献是经过次数<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\geq k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≥</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>的点数，求所有路径的贡献和</p>
<p>此外，还有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span>个特殊点，路径不得经过这些特殊点</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo separator="true">,</mo><mi>t</mi><mo separator="true">,</mo><mi>k</mi><mo>≤</mo><mn>3</mn><mn>0</mn></mrow><annotation encoding="application/x-tex">n,t,k\leq 30</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span><span class="mpunct">,</span><span class="mord mathit">t</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">≤</span><span class="mord mathrm">3</span><span class="mord mathrm">0</span></span></span></span></p>
<h3 id="题解"><a class="markdownIt-Anchor" href="#题解"></a> 题解</h3>
<p>显然可以对每个位置分别考虑贡献。即：对于每个位置，统计有多少路径经过这个位置<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\geq k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≥</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>次</p>
<p>我们需要dp求出以下三个东西</p>
<ul>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{x,y,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mord mathit">t</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>：从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span>出发，走<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span></span></span></span>步后第一次到达<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>的路径数</li>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>g</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{x,y,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mord mathit">t</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>：从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>出发，走<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span></span></span></span>步后第一次回到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>的路径数</li>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">h_{x,y,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mord mathit">t</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>：从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>出发，随便走<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span></span></span></span>步的路径数</li>
</ul>
<p>dp很简单，而且都是一样的，复制三遍改改细节就行</p>
<p>然后，我们设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub><mo separator="true">,</mo><msub><mi>G</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub><mo separator="true">,</mo><msub><mi>H</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">F_{x,y},G_{x,y},H_{x,y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">G</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.08125em;">H</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.08125em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>分别为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub><mo separator="true">,</mo><msub><mi>g</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub><mo separator="true">,</mo><msub><mi>h</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{x,y,t},g_{x,y,t},h_{x,y,t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mord mathit">t</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mord mathit">t</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mord mathit">t</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>的普通型生成函数，那么，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>的答案显然就是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub><mo>∗</mo><msubsup><mi>G</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>∗</mo><msub><mi>H</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub><mo>[</mo><msup><mi>z</mi><mi>t</mi></msup><mo>]</mo></mrow><annotation encoding="application/x-tex">F_{x,y}*G_{x,y}^{k-1}*H_{x,y}[z^t]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.849108em;"></span><span class="strut bottom" style="height:1.232216em;vertical-align:-0.383108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">∗</span><span class="mord"><span class="mord mathit">G</span><span class="vlist"><span style="top:0.24700000000000003em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">∗</span><span class="mord"><span class="mord mathit" style="margin-right:0.08125em;">H</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.08125em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">t</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">]</span></span></span></span></p>
<p>复杂度<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>t</mi><mn>4</mn></msup><mi>k</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(t^4k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">t</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>，时限比较卡，需要卡常。dp需要模优化，求<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>G</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">G_{x,y}^{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.849108em;"></span><span class="strut bottom" style="height:1.232216em;vertical-align:-0.383108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">G</span><span class="vlist"><span style="top:0.24700000000000003em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>的时候还要用多项式快速幂优化</p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br><span class="line">124</span><br><span class="line">125</span><br><span class="line">126</span><br><span class="line">127</span><br><span class="line">128</span><br><span class="line">129</span><br><span class="line">130</span><br><span class="line">131</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> LL;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> ha=<span class="number">998244353</span>,N=<span class="number">65</span>,B=<span class="number">32</span>;</span><br><span class="line"><span class="keyword">int</span> f[N][N][N]; <span class="comment">//f[x][y][t]: 从(0,0)出发，走了t步后第一次到达(x,y)的方案数</span></span><br><span class="line"><span class="keyword">int</span> g[N][N][N]; <span class="comment">//g[x][y][t]: 从(x,y)出发，走了t步后第一次回到(x,y)的方案数</span></span><br><span class="line"><span class="keyword">int</span> h[N][N][N]; <span class="comment">//h[x][y][t]: 从(x,y)出发，随便乱走t步的方案数</span></span><br><span class="line"><span class="keyword">int</span> t,n,k,bad[N][N];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> &amp;x,<span class="keyword">const</span> <span class="keyword">int</span> &amp;y)</span></span>&#123;x=(x+y&gt;=ha)?(x+y-ha):(x+y);&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">gao_f</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> x=-t;x&lt;=t;x++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> y=<span class="built_in">abs</span>(x)-t;y&lt;=t-<span class="built_in">abs</span>(x);y++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span>(bad[B+x][B+y]) <span class="keyword">continue</span>;</span><br><span class="line">            <span class="keyword">if</span>(x==<span class="number">0</span>&amp;&amp;y==<span class="number">0</span>)&#123;f[B+x][B+y][<span class="number">0</span>]=<span class="number">1</span>;<span class="keyword">continue</span>;&#125;</span><br><span class="line">            <span class="keyword">static</span> <span class="keyword">int</span> dp[<span class="number">2</span>][N][N];</span><br><span class="line">            <span class="built_in">memset</span>(dp[<span class="number">0</span>],<span class="number">0</span>,<span class="keyword">sizeof</span>(dp[<span class="number">0</span>]));</span><br><span class="line">            <span class="keyword">int</span> cur=<span class="number">0</span>;dp[cur][B][B]=<span class="number">1</span>;</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=t;i++,cur^=<span class="number">1</span>)</span><br><span class="line">            &#123;</span><br><span class="line">                <span class="built_in">memset</span>(dp[cur^<span class="number">1</span>],<span class="number">0</span>,<span class="keyword">sizeof</span>(dp[cur^<span class="number">1</span>]));</span><br><span class="line">                <span class="keyword">for</span>(<span class="keyword">int</span> j=-t;j&lt;=t;j++)</span><br><span class="line">                    <span class="keyword">for</span>(<span class="keyword">int</span> k=<span class="built_in">abs</span>(j)-t;k&lt;=t-<span class="built_in">abs</span>(j);k++)</span><br><span class="line">                    &#123;</span><br><span class="line">                        <span class="keyword">if</span>(bad[B+j][B+k]) <span class="keyword">continue</span>;</span><br><span class="line">                        <span class="keyword">int</span> tmp=dp[cur][B+j<span class="number">-1</span>][B+k];</span><br><span class="line">                        add(tmp,dp[cur][B+j+<span class="number">1</span>][B+k]);</span><br><span class="line">                        add(tmp,dp[cur][B+j][B+k<span class="number">-1</span>]);</span><br><span class="line">                        add(tmp,dp[cur][B+j][B+k+<span class="number">1</span>]);</span><br><span class="line">                        <span class="keyword">if</span>(j==x&amp;&amp;k==y) f[B+x][B+y][i]=tmp;</span><br><span class="line">                        <span class="keyword">else</span> dp[cur^<span class="number">1</span>][B+j][B+k]=tmp;</span><br><span class="line">                    &#125;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">gao_g</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> x=-t;x&lt;=t;x++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> y=<span class="built_in">abs</span>(x)-t;y&lt;=t-<span class="built_in">abs</span>(x);y++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span>(bad[B+x][B+y]) <span class="keyword">continue</span>;</span><br><span class="line">            <span class="keyword">static</span> <span class="keyword">int</span> dp[<span class="number">2</span>][N][N];</span><br><span class="line">            <span class="built_in">memset</span>(dp[<span class="number">0</span>],<span class="number">0</span>,<span class="keyword">sizeof</span>(dp[<span class="number">0</span>]));</span><br><span class="line">            <span class="keyword">int</span> cur=<span class="number">0</span>;dp[cur][B+x][B+y]=<span class="number">1</span>;</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=t;i++,cur^=<span class="number">1</span>)</span><br><span class="line">            &#123;</span><br><span class="line">                <span class="built_in">memset</span>(dp[cur^<span class="number">1</span>],<span class="number">0</span>,<span class="keyword">sizeof</span>(dp[cur^<span class="number">1</span>]));</span><br><span class="line">                <span class="keyword">for</span>(<span class="keyword">int</span> j=-t;j&lt;=t;j++)</span><br><span class="line">                    <span class="keyword">for</span>(<span class="keyword">int</span> k=<span class="built_in">abs</span>(j)-t;k&lt;=t-<span class="built_in">abs</span>(j);k++)</span><br><span class="line">                    &#123;</span><br><span class="line">                        <span class="keyword">if</span>(bad[B+j][B+k]) <span class="keyword">continue</span>;</span><br><span class="line">                        <span class="keyword">int</span> tmp=dp[cur][B+j<span class="number">-1</span>][B+k];</span><br><span class="line">                        add(tmp,dp[cur][B+j+<span class="number">1</span>][B+k]);</span><br><span class="line">                        add(tmp,dp[cur][B+j][B+k<span class="number">-1</span>]);</span><br><span class="line">                        add(tmp,dp[cur][B+j][B+k+<span class="number">1</span>]);</span><br><span class="line">                        <span class="keyword">if</span>(j==x&amp;&amp;k==y) g[B+x][B+y][i]=tmp;</span><br><span class="line">                        <span class="keyword">else</span> dp[cur^<span class="number">1</span>][B+j][B+k]=tmp;</span><br><span class="line">                    &#125;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">gao_h</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> x=-t;x&lt;=t;x++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> y=<span class="built_in">abs</span>(x)-t;y&lt;=t-<span class="built_in">abs</span>(x);y++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span>(bad[B+x][B+y]) <span class="keyword">continue</span>;</span><br><span class="line">            <span class="keyword">static</span> <span class="keyword">int</span> dp[<span class="number">2</span>][N][N];<span class="keyword">int</span> cur=<span class="number">0</span>;</span><br><span class="line">            <span class="built_in">memset</span>(dp[<span class="number">0</span>],<span class="number">0</span>,<span class="keyword">sizeof</span>(dp[<span class="number">0</span>]));</span><br><span class="line">            dp[<span class="number">0</span>][B+x][B+y]=h[B+x][B+y][<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=t;i++,cur^=<span class="number">1</span>)</span><br><span class="line">            &#123;</span><br><span class="line">                <span class="built_in">memset</span>(dp[cur^<span class="number">1</span>],<span class="number">0</span>,<span class="keyword">sizeof</span>(dp[cur^<span class="number">1</span>]));</span><br><span class="line">                <span class="keyword">for</span>(<span class="keyword">int</span> j=-t;j&lt;=t;j++)</span><br><span class="line">                    <span class="keyword">for</span>(<span class="keyword">int</span> k=<span class="built_in">abs</span>(j)-t;k&lt;=t-<span class="built_in">abs</span>(j);k++)</span><br><span class="line">                    &#123;</span><br><span class="line">                        <span class="keyword">if</span>(bad[B+j][B+k]) <span class="keyword">continue</span>;</span><br><span class="line">                        <span class="keyword">int</span> tmp=dp[cur][B+j<span class="number">-1</span>][B+k];</span><br><span class="line">                        add(tmp,dp[cur][B+j+<span class="number">1</span>][B+k]);</span><br><span class="line">                        add(tmp,dp[cur][B+j][B+k<span class="number">-1</span>]);</span><br><span class="line">                        add(tmp,dp[cur][B+j][B+k+<span class="number">1</span>]);</span><br><span class="line">                        add(h[B+x][B+y][i],tmp);</span><br><span class="line">                        dp[cur^<span class="number">1</span>][B+j][B+k]=tmp;</span><br><span class="line">                    &#125;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">polymul</span><span class="params">(<span class="keyword">int</span> *a,<span class="keyword">int</span> *b,<span class="keyword">int</span> *res)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">static</span> <span class="keyword">int</span> tmp[N];</span><br><span class="line">    <span class="built_in">memset</span>(tmp,<span class="number">0</span>,<span class="keyword">sizeof</span>(<span class="keyword">int</span>)*(t+<span class="number">1</span>));</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;=t;i++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> j=<span class="number">0</span>;j&lt;=t-i;j++)</span><br><span class="line">            tmp[i+j]=(tmp[i+j]+(LL)a[i]*b[j])%ha;</span><br><span class="line">    <span class="built_in">memcpy</span>(res,tmp,<span class="keyword">sizeof</span>(<span class="keyword">int</span>)*(t+<span class="number">1</span>));</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">polypow</span><span class="params">(<span class="keyword">int</span> *a,<span class="keyword">int</span> b,<span class="keyword">int</span> *res)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="built_in">memset</span>(res,<span class="number">0</span>,<span class="keyword">sizeof</span>(<span class="keyword">int</span>)*(t+<span class="number">1</span>));res[<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line">    <span class="keyword">for</span>(;b;b&gt;&gt;=<span class="number">1</span>,polymul(a,a,a))</span><br><span class="line">        <span class="keyword">if</span>(b&amp;<span class="number">1</span>) polymul(res,a,res);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="built_in">cin</span>&gt;&gt;t&gt;&gt;n&gt;&gt;k;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>,x,y;i&lt;=n;i++)</span><br><span class="line">        <span class="built_in">cin</span>&gt;&gt;x&gt;&gt;y,bad[B+x][B+y]=<span class="number">1</span>;</span><br><span class="line">    gao_f();gao_g();gao_h();</span><br><span class="line">    <span class="keyword">int</span> ans=<span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> x=-t;x&lt;=t;x++)</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> y=<span class="built_in">abs</span>(x)-t;y&lt;=t-<span class="built_in">abs</span>(x);y++)</span><br><span class="line">        &#123;</span><br><span class="line">            <span class="keyword">if</span>(bad[B+x][B+y]) <span class="keyword">continue</span>;</span><br><span class="line">            <span class="keyword">static</span> <span class="keyword">int</span> poly[N];</span><br><span class="line">            polypow(g[B+x][B+y],k<span class="number">-1</span>,poly);</span><br><span class="line">            polymul(poly,f[B+x][B+y],poly);</span><br><span class="line">            polymul(poly,h[B+x][B+y],poly);</span><br><span class="line">            add(ans,poly[t]);</span><br><span class="line">        &#125;</span><br><span class="line">    <span class="built_in">cout</span>&lt;&lt;ans&lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div>
      
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